@MAK: If you study PDEs, you should know about "weak solutions". To your first question: For any test function $\varphi$ with support away from the origin, you will have $\delta(\varphi) = 0$, hence $(LE)(\varphi) = 0$.
–
VoboJan 3 '13 at 11:42

1 Answer
1

You should not forget the meaning of $\delta$. It is a linear functional on (for instance) the compactly supported smooth functions on $\mathbf R^d$. That is, $LE = \delta$ means for all such $\psi$ that
$$\langle LE, \psi \rangle = \langle \delta, \psi \rangle = \psi(0).$$
Also, the support of $\delta$ is $\{0\}$ as can be seen above. Hence, if we test against $\psi$ which vanishes in $0$ we get
$$\langle LE, \psi \rangle = 0.$$
Obtaining that $LE = 0$ outside the support of $\delta$, that is, outside $\{0\}$.

Concerning your second question, the question reduces to "which class of functions do you test again", as recall that, distributions are functionals that have a certain domain. If $k \neq 0$, this only works on classes of test functions vanishing in $0$.